How to insert keyframe (a modifier property) to all selected objects?
I originally thought you were using a driver for the angle property.  I don't know why I brain-farted like that.  But-- you should use a driver for shared properties like this, so that's the first thing we'll do.
Create an empty.  Right click in its X rotation field (for example) and select &quot;copy as new driver.&quot;
Select one object.  Right click on the angle field and select &quot;paste as new driver.&quot;
Now test it out by rotating your empty (in the X axis.)  You'll see your modifier's angle changes with the rotation of the empty.
But we want that on the rest of the objects as well.  Select the rest of the objects, with the first as the active selection, and link modifiers (ctrl l -&gt; modifiers.) The modifier on the actively selected object, including any driven values, will be copied to all selected objects.
Try rotating your empty again.  You'll see that the angle changes, in unison, for all of the objects to which you copied the driven modifier.
You can drive from something other than rotation, something other than an empty, if you'd prefer.  But if you want everything acting in unison, it's nice to have a single thing to animate.  Frequent choices for driver targets are custom properties and bone transforms (the latter, because it vastly simplifies combining animations into single NLA strips.)